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### Course: AP®︎ Calculus BC (2017 edition)>Unit 4

Lesson 10: Optional videos

# Proof: The derivative of 𝑒ˣ is 𝑒ˣ

Proving that the derivative of 𝑒ˣ is 𝑒ˣ using the definition of the derivative as a limit, and the definition of 𝑒 as a limit.

## Want to join the conversation?

• Can anyone prove that lim n-->0 (1+n)^1/n = e ? I remember learning the first form, but not this one, so knowing that the second form is true will make this proof much easier to understand.
• First form:
Lim n->inf (1+1/n)^n
Subst: n=1/m
Lim 1/m -> inf (1+m)^1/m
Change to equivalent limit (if 1/m -> inf, then thats because m -> 0):
Lim m->0 (1+m)^1/m
• what limit property did sal use at to replace what was inside the natural log with the limit of 1 plus n to the one over n, as n approaches 0?
• Putting the limit inside of the natural log wont change it because the only n is inside of the natural log. So if you put it outside or inside, it will still have the same effect on n.
(1 vote)
• what limit property did sal use at to move the limit into the ln( ) parentheses:
lim h-->0 {ln(1+n)^1/n } = ln { lim h-->0 (1+n)^1/n }
• Say I have the expression lim x->3 (ln(x)). Clearly, this is just equal to ln(3), regardless of the fact that it's inside a logarithm. It's just what the expression equals when x=3. So, lim x->3 (ln(x)) = ln(3). Further, ln(lim x->3 (x)) = ln(3), which I think is clear enough to not warrant a lengthy explanation (let me know if not, though). In this case, it doesn't matter that x is inside a logarithm, since the only part of the expression being affected by the limit is inside the logarithm, and this also applies to the more complex example in the video.
• At ,why can we factor out e to the x? I mean it can only be factored when it is a constant right?
• I'm also a bit confused. I think it's because he is taking the limit as dx approaches 0. So x itself won't change as you change dx around it. So you can treat it as a constant and use the constant multiplication rule to bring it out.

See here for something of an explanation: https://www.youtube.com/watch?v=m2MIpDrF7Es&feature=youtu.be&t=345
• L'hopital rule can be used to find the limit
• But that requires already knowing the derivative of e^x.
• What is the derrivative of e to the power 4x
• The answer would be 4e^4x. You just need to use the rule.
• "now you might be getting that tingly feeling"